We give necessary and sufficient conditions for the existence of a decomposition of the complete graph into stars which admits either a cyclic or a rotational automorphism.

This paper deals with combinatorial aspects of designs for two-way elimination of heterogeneity for making all possible paired comparisons of treatments belonging to two disjoint sets of treatments. Balanced bipartite row-column (BBPRC) designs have been defined which estimate all the elementary contrasts involving two treatments one from each of the two disjoint sets with the same variance. General efficiency balanced row-column designs (GEBRC) are also defined. Some general methods of construction of BBPRC designs have been given using the techniques of reinforcement, deletion (addition) of column or row structures, merging of treatments, balanced bipartite block (BBPB) designs, juxtaposition, etc. Some methods of construction give GEBRC designs also.

A critical set in a Latin square of order \(n\) is a set of entries in a Latin square which can be embedded in precisely one Latin square of order \(n\). Also, if any element of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order \(n\). In this paper, we find smallest weak and smallest totally weak critical sets for all the Latin squares of orders six and seven. Moreover, we computationally prove that there is no (totally) weak critical set in the back circulant Latin square of order five and we find a totally weak critical set of size seven in the other main class of Latin squares of order five.

In this paper, we give the following labelings:

1. Elegant labelings of triangular snakes \(\Delta_{n}\) , \(n \equiv 0,1,2 \mod 4\).
2. Near-elegant labeling of triangular snakes \(\Delta_{n}\) when \(n \equiv 3 \mod 4\), which are not elegant.
3. Elegant and near-elegant labelings of some of the theta graphs \(\theta_{n,n}\) when \(n = 1, 2, 3\).
4. Harmonious labelings of helms \(H_n\) when \(n\) is even.

A linear \([n,k,d]_q\) code \(C\) is called NMDS if \(d(C) = n – k\) and \(d(C^{\perp}) = k\). In this paper, the classification of the \([n,3,n-k]_q\) NMDS codes is given for \(q = 7,8,9\). It has been found using the correspondence between \([n,3,n-k]_q\) NMDS codes and \((n,3)\)-arcs of \(\mathrm{PG}(2,q)\).

A path in a digraph is antidirected if the two adjacent edges of the path have opposing orientations. In this paper, we give a necessary and sufficient condition for the edges of the complete symmetric graph to be decomposed into isomorphic antidirected paths.

The aim of this note is to provide a programme for the Computer Algebra package MAGMA, which is suitable to decode one-point Goppa codes defined from Hermitian curves.

In this article, the intersection problem for twin bowtie and near bowtie systems is completely solved.